q-Leibniz Rule?

Question

The famous Leibniz rule: $(uv)' = u'v+uv'$ can be generalized for any order derivative using the following formula $$(uv)^{(n)} = \sum_{k=0}^n{n\choose k}u^{(n-k)}v^{(k)}$$ This is a known formula which is relatively easy to prove. More on it can be read here. I was wondering if one could create a q-analog using the q-Binomial coefficient (also known as the Gauss binomial coefficient) as demonstrated below $$(uv)^{(n)}_q = \sum_{k=0}^n{n\choose k}_qu^{(n-k)}v^{(k)}$$ A few examples: $$(uv)_q' = u'v+uv'$$ $$(uv)_q'' = u''v+(1+q)u'v'+uv''$$ $$(uv)_q''' = u'''v+(1+q+q^2)u''v'+(1+q+q^2)u'v''+uv'''$$ It satisfies the requirement for the definition of a q-analog as $q\rightarrow 1$ is the regular Leibniz rule. Just to clarify, I'm aware of the higher-order analogs of the q-derivative, and that they look similar in concept to the formula above, however my focus is on the general Leibniz rule for any $u$ and $v$. This extension/analog seems pretty natural and can be used to extend the concept of derivations and their higher-order forms. I was just wondering if this is actually a useful or interesting q-analog, as well as if anyone has anything they'd like to contribute. Please feel free to share whatever insights, thoughts, formulas, references, etc you like